Nonexistence of star-supported spline bases

We consider polynomial spline spaces S-d(r)(Delta) of degree d and smoothne ss r defined on triangulations. It is known that for d greater than or equa l to 3r + 2, S-d(r)(Delta) possesses a basis of star-supported splines, i.e ., splines whose supports are at most the set of triangles surrounding a ve rtex. Here we extend the theory by showing that for all d less than or equa l to 3r + 1, there exist triangulations for which no such bases exist.